Theorem mexval 35496

Description: The set of expressions, which are pairs whose first element is a typecode, and whose second element is a raw expression. (Contributed by Mario Carneiro, 18-Jul-2016.)

Hypotheses

Ref	Expression
mexval.k	⊢ 𝐾 = (mTC‘𝑇)
mexval.e	⊢ 𝐸 = (mEx‘𝑇)
mexval.r	⊢ 𝑅 = (mREx‘𝑇)

Assertion

Ref	Expression
mexval	⊢ 𝐸 = (𝐾 × 𝑅)

Proof of Theorem mexval

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mexval.e	. 2 ⊢ 𝐸 = (mEx‘𝑇)
2		fveq2 6861	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mTC‘𝑡) = (mTC‘𝑇))
3		mexval.k	. . . . . 6 ⊢ 𝐾 = (mTC‘𝑇)
4	2, 3	eqtr4di 2783	. . . . 5 ⊢ (𝑡 = 𝑇 → (mTC‘𝑡) = 𝐾)
5		fveq2 6861	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mREx‘𝑡) = (mREx‘𝑇))
6		mexval.r	. . . . . 6 ⊢ 𝑅 = (mREx‘𝑇)
7	5, 6	eqtr4di 2783	. . . . 5 ⊢ (𝑡 = 𝑇 → (mREx‘𝑡) = 𝑅)
8	4, 7	xpeq12d 5672	. . . 4 ⊢ (𝑡 = 𝑇 → ((mTC‘𝑡) × (mREx‘𝑡)) = (𝐾 × 𝑅))
9		df-mex 35481	. . . 4 ⊢ mEx = (𝑡 ∈ V ↦ ((mTC‘𝑡) × (mREx‘𝑡)))
10		fvex 6874	. . . . 5 ⊢ (mTC‘𝑡) ∈ V
11		fvex 6874	. . . . 5 ⊢ (mREx‘𝑡) ∈ V
12	10, 11	xpex 7732	. . . 4 ⊢ ((mTC‘𝑡) × (mREx‘𝑡)) ∈ V
13	8, 9, 12	fvmpt3i 6976	. . 3 ⊢ (𝑇 ∈ V → (mEx‘𝑇) = (𝐾 × 𝑅))
14		xp0 6134	. . . . 5 ⊢ (𝐾 × ∅) = ∅
15	14	eqcomi 2739	. . . 4 ⊢ ∅ = (𝐾 × ∅)
16		fvprc 6853	. . . 4 ⊢ (¬ 𝑇 ∈ V → (mEx‘𝑇) = ∅)
17		fvprc 6853	. . . . . 6 ⊢ (¬ 𝑇 ∈ V → (mREx‘𝑇) = ∅)
18	6, 17	eqtrid 2777	. . . . 5 ⊢ (¬ 𝑇 ∈ V → 𝑅 = ∅)
19	18	xpeq2d 5671	. . . 4 ⊢ (¬ 𝑇 ∈ V → (𝐾 × 𝑅) = (𝐾 × ∅))
20	15, 16, 19	3eqtr4a 2791	. . 3 ⊢ (¬ 𝑇 ∈ V → (mEx‘𝑇) = (𝐾 × 𝑅))
21	13, 20	pm2.61i 182	. 2 ⊢ (mEx‘𝑇) = (𝐾 × 𝑅)
22	1, 21	eqtri 2753	1 ⊢ 𝐸 = (𝐾 × 𝑅)

Syntax hints:  ¬ wn 3   = wceq 1540   ∈ wcel 2109  Vcvv 3450  ∅c0 4299   × cxp 5639  ‘cfv 6514  mTCcmtc 35458  mRExcmrex 35460  mExcmex 35461

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-iota 6467  df-fun 6516  df-fv 6522  df-mex 35481

This theorem is referenced by:  mexval2  35497  msubff  35524  msubco  35525  msubff1  35550  mvhf  35552  msubvrs  35554